If \(\eta > 0\) , we apply Lemma REF  to obtain a graph \(G^{\prime }\)  and an integer \(\Delta \)  such that \(G^{\prime } - X\)  has at most \(X^{\mathcal {O}_\eta (1)}\)  connected components and \(\textnormal {\textsc {fvs}}(G^{\prime }) + \Delta = \textnormal {\textsc {fvs}}(G)\) .
Observe that \(G^{\prime }\)  and \(\Delta \)  can be obtained in polynomial time for fixed \(\eta \) .
We will now extend the modulator \(X\)  to a modulator \(X^{\prime }\)  such that \(G^{\prime } - X^{\prime }\)  has \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(G^{\prime } - X^{\prime }) < \eta \) .
Consider each connected component \(C\)  of \(G^{\prime } - X\)  for which \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(C) = \eta \) .
By definition, for each such component \(C\)  there must exist a vertex \(v \in V(C)\)  such that \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(C - v) < \eta \) .
We can find such a vertex by trying all options for \(v\)  and determining \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(C - v)\) .
This can be done in polynomial time for constant \(\eta \)  since determining elimination distances to minor-closed graph classes is fixed parameter tractable in the target value [1]}.
By adding this vertex to \(X\)  for each considered component \(C\) , we obtain a set \(X^{\prime }\)  whose size is still bounded by a polynomial in \(X\)  for constant \(\eta \)  since the number of components to consider is polynomial in \(X\) .
Since each connected component \(C\)  of \(G^{\prime } - X^{\prime }\)  has \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(C) < \eta \) , it follows that \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(G^{\prime } - X^{\prime }) < \eta \) .
Then by induction, the Feedback Vertex Set problem on \(G^{\prime }\)  admits a polynomial kernel in the size of \(X^{\prime }\) , which is a modulator to a graph with elimination distance to a forest at most \(\eta - 1\) .
Since \(\textnormal {\textsc {fvs}}(G^{\prime }) + \Delta = \textnormal {\textsc {fvs}}(G)\)  by Lemma REF , we know that \(G^{\prime }\)  has a feedback vertex set of size at most \(\ell \)  if and only if \(G\)  has a feedback vertex set of size at most \(\ell + \Delta \) .
This concludes the induction step.
